To find the equilibrium price, we need to set the supply \( O \) equal to the demand \( D \). The given functions are:
\[ O = 50P - 300 \] \[ D = 150 - 10P \]
a) Calculate the equilibrium price.
To find equilibrium, set \( O \) equal to \( D \):
\[ 50P - 300 = 150 - 10P \]
Now, let's solve for \( P \):
- Add \( 10P \) to both sides:
\[ 50P + 10P - 300 = 150 \]
\[ 60P - 300 = 150 \]
- Add \( 300 \) to both sides:
\[ 60P = 450 \]
- Divide by \( 60 \):
\[ P = \frac{450}{60} = 7.5 \]
Thus, the equilibrium price \( P^* = 7.5 \).
What happens if the price is higher or lower than $7.5:
-
If the price is higher than $7.5 (for example, at \( P = 10 \)):
- Calculate supply:
\[ O = 50(10) - 300 = 500 - 300 = 200 \]
- Calculate demand:
\[ D = 150 - 10(10) = 150 - 100 = 50 \]
- In this case, supply (200) exceeds demand (50), leading to a surplus of goods. Producers will lower their prices to clear this surplus, moving the price down toward equilibrium.
-
If the price is lower than $7.5 (for example, at \( P = 5 \)):
- Calculate supply:
\[ O = 50(5) - 300 = 250 - 300 = -50 \] (indicating no supply)
- Calculate demand:
\[ D = 150 - 10(5) = 150 - 50 = 100 \]
- In this case, demand (100) exceeds supply (0), indicating a shortage of goods. Consumers will compete for the limited supply, driving the price up toward equilibrium.
b) Calculate the demand and supply for the prices of $5 and $10.
At \( P = 5 \):
- Supply:
\[ O = 50(5) - 300 = 250 - 300 = -50 \quad \text{(no supply)} \]
- Demand:
\[ D = 150 - 10(5) = 150 - 50 = 100 \]
At \( P = 10 \):
- Supply:
\[ O = 50(10) - 300 = 500 - 300 = 200 \]
- Demand:
\[ D = 150 - 10(10) = 150 - 100 = 50 \]
Summary of Results:
-
At \( P = 5 \):
- Supply = 0 (no supply)
- Demand = 100
-
At \( P = 10 \):
- Supply = 200
- Demand = 50
The equilibrium price is \( P = 7.5 \), with implications of surplus at higher prices and shortages at lower prices.
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